A weak criterion for vector-valued holomorphy

We study functions f of one complex variable with values in a complete locally convex space E. In [13] we had shown that for f to be holomorphic it suffices that it is locally bounded and that there is a separating subset H of E' such that x' o f is holomorphic for each functional x' is an element of H. We give here a new and considerably shorter proof of this result, and we extend it to a weak criterion for holomorphic continuability.